Example 2.7 
Suppose that the noise random variable \(\xi \)  satisfies the sub-Gaussian assumption of the form
\(\Vert \xi \Vert _{\psi _2} \lesssim \Vert \xi \Vert _{L_2}\) . Then, by standard sub-Gaussian maximum inequalities (e.g., [1]}), we have
\(\Vert \max _i \xi _i\Vert _{L_2}^2 \lesssim \Vert \xi \Vert _{L_2}^2\log n = R(\omega ^*)\log n\) 
and hence the upper bound (REF ) simplifies to
\(\frac{\operatorname{\mathbb {E}}\xi ^2X^\mathsf {T}\widehat{\Sigma }_{\lambda }^{-1}X}{n}\lesssim \frac{dR(\omega ^{*}_{b}) \log n}{n}.\) 

